Python: Make an Operator update UI after executed
You can use static members (class variables) to store information in the operator.

Here's an example that adds a panel and an operator. The operator increases a class variable to reflect how often it has been called. This number is drawn in a panel in the Scene tab of the Properties Edtior:

import bpy

class HelloWorldPanel(bpy.types.Panel):
    """Creates a Panel in the Object properties window"""
    bl_label = "Hello World Panel"
    bl_idname = "OBJECT_PT_hello"
    bl_space_type = 'PROPERTIES'
    bl_region_type = 'WINDOW'
    bl_context = "scene"

    def draw(self, context):
        layout = self.layout

        layout.operator(SimpleOperator.bl_idname)
        layout.label(str(SimpleOperator.run))


class SimpleOperator(bpy.types.Operator):
    """Tooltip"""
    bl_idname = "object.simple_operator"
    bl_label = "Simple Object Operator"

    run = 0

    def execute(self, context):
        self.__class__.run += 1
        self.report({'INFO'}, str(self.__class__.run))
        return {'FINISHED'}


def register():
    bpy.utils.register_module(__name__)


def unregister():
    bpy.utils.unregister_module(__name__)


if __name__ == "__main__":
    register()